### Franc-carreau or Fair-square

Franc-carreau is a simple game of chance, like the roll-a-penny game often seen at fairs and fêtes. A coin is tossed or rolled down a wooden chute onto a large board ruled into square segments. If the player’s coin lands completely within a square, he or she wins a coin of equal value. If the coin crosses a dividing line, it is lost.

The playing board for Franc-Carreau is shown above, together with a winning coin (red) contained within a square and a loosing one (blue) crossing a line. As the precise translation of franc-carreau appears uncertain, the name “fair square” would seem appropriate.

The question is: What size should the coin be to ensure a 50% chance of winning?

We assume that each square has sides of unit length. A coin of radius r just touches the edge of the unit square if its centre lies on an inner square of side-length (1 – 2 r). To win, the coin centre must land within the smaller square. Clearly, the smaller r is, the greater the chance of winning.

We assume that all points within the unit square are equally likely, so the probability of the centre being within the inner square is ( 1 – 2 r ) 2 . For a 50% chance of victory, we have

( 1 – 2 r ) 2 = ½         or         4 r 2 – 4 r + ½ = 0

This quadratic equation has two roots, one greater than ½ and one less than ½. Only the latter is of interest:

r = ( 2 – 2 ) / 4 ≈ 0.146 1 / 7 .

So, if the coin radius is about one-seventh the side-length of the square, there is a roughly 50% chance of winning. In many fair-grounds, coins or tokens are markedly larger than this!

Buffon and his Needle

When Georges-Louis Leclerc, Compte de Buffon (1707 – 1788) was just 20, he discovered the binomial theorem (for integer exponents) independently of Newton. He was greatly interested in probability, approaching that topic – like many other scholars – from its connections with gambling. He considered the question: can one be sure of winning in roulette by repeatedly doubling one’s wager until one is in profit? This is related to the notorious Gambler’s Ruin. Buffon’s 1733 Mémoire sur le jeu de franc-carreau earned him election to the Royal Academy of Sciences. He published extensively, including 36 volumes of an encyclopedia on the natural sciences.

Laplace discussed Buffon’s work in his Théorie analytique des probabilités (1812). Buffon was the first mathematician to deal with random trials. This was in his work on franc-carreau and similar games, which was a clever combination of ideas from two distinct areas, geometry and probability. It can be considered the origin of geometric probability. His study of franc-carreau led immediately to the problem now called Buffon’s Needle [See article on Buffon’s Needle on this blog]

For an interesting account of Buffon’s life and work, see the article Georges-Louis Leclerc Buffon (1707–1788), by Heinz Klaus Strick, on the Mathematics-in-Europe web site.

Sources

Heinz Klaus Strick: Georges-Louis Leclerc Buffon  at Mathematics-in-Europe